Determine the Hausdorff structure of quantum extended objects

Determine how the Hausdorff dimension and geometric structure of a quantum extended object such as a $p$-brane change when its classical point-like description is replaced by an extended quantum configuration.

Background

The paper derives a scale-dependent Hausdorff dimension for quantum particle trajectories and argues that quantum strings acquire a fractal, effectively three-dimensional world-sheet because of quantum fluctuations.

It then raises the corresponding problem for general extended objects, including pp-branes. The string case is explored as an instructive example, but the paper explicitly acknowledges that the general question has not been fully answered.

References

At this point one may be tempted to ask what happens if the classical object is no longer a point-like object but an extended one, for instance a $p$-brane. Even if we do not have a full answer to this question, it is instructive to explore what happens in the case $p=1$, namely the string.

— Condensates: A Bridge to Quantum Gravity  (2609.28789 - Nicolini, 23 Sep 2026) in Section 4, “The Spacetime Condensate”